# sums of powers

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##### 3: 27.2 Functions
27.2.6 $\phi_{k}\left(n\right)=\sum_{\left(m,n\right)=1}m^{k},$
the sum of the $k$th powers of the positive integers $m\leq n$ that are relatively prime to $n$.
27.2.7 $\phi\left(n\right)=\phi_{0}\left(n\right).$
27.2.10 $\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}n}d^{\alpha},$
is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. …
##### 4: 27.3 Multiplicative Properties
27.3.7 $\sigma_{\alpha}\left(m\right)\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}% \left(m,n\right)}d^{\alpha}\sigma_{\alpha}\left(\frac{mn}{d^{2}}\right),$
##### 5: 27.7 Lambert Series as Generating Functions
If $|x|<1$, then the quotient $x^{n}/(1-x^{n})$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: …
27.7.5 $\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sigma_{% \alpha}\left(n\right)x^{n},$
##### 7: 24.19 Methods of Computation
• Tanner and Wagstaff (1987) derives a congruence $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

• ##### 8: Bibliography
• T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
• ##### 9: 27.4 Euler Products and Dirichlet Series
27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$,
##### 10: 27.11 Asymptotic Formulas: Partial Sums
27.11.4 $\sum_{n\leq x}\sigma_{1}\left(n\right)=\frac{{\pi}^{2}}{12}x^{2}+O\left(x\ln x% \right).$
27.11.5 $\sum_{n\leq x}\sigma_{\alpha}\left(n\right)=\frac{\zeta\left(\alpha+1\right)}{% \alpha+1}x^{\alpha+1}+O\left(x^{\beta}\right),$ $\alpha>0$, $\alpha\neq 1$, $\beta=\max(1,\alpha)$.